3 ms·

Fyi, "3/3 = 1" is true as well, which you seem to be indicating is not the case. 3/3 does not approach 1; it is exactly 1. In fact, the idea of "approaches" can

by euank 12y ago

Fyi, "3/3 = 1" is true as well, which you seem to be indicating is not the case. 3/3 does not approach 1; it is exactly 1. In fact, the idea of "approaches" cannot happen with just rational numbers. You need a limit in order for the word "approaches" to be correct. I think you need to read up on math terminology a little (and just math in general) if you want to try and discuss this... though there's no point discussing this because that implies there's more than one side. In the case of .99.., that could be expressed using a limit and thus you could say it approaches 1, though it's much more accurate to say that it equals 1.

pdonis 12y ago
> In fact, the idea of "approaches" cannot happen with just rational numbers. Yes, it can. The sequence 0.9, 0.99, 0.999, ... is entirely composed of rational numbers, and its limit is also a rational number. The issue with rational numbers is that there are sequences which are composed entirely of rational numbers but do not have limits that are rational numbers. An example is 1.4, 1.41, 1.414, ..., i.e., successive truncations of the decimal expansion of sqrt(2). The limit of the sequence is sqrt(2) itself, which is irrational. So in that particular case, the concept of "taking the limit of a sequence" can't be applied within the rational numbers. But that doesn't mean it can't be applied at all; you just need to restrict to sequences that have rational limits.
- ColinWright 12y ago
  Just a thought ... euank> In fact, the idea of "approaches" cannot euank> happen with just rational numbers. pdonis> Yes, it can. The sequence 0.9, 0.99, 0.999, ... pdonis> is entirely composed of rational numbers, and its pdonis> limit is also a rational number. Everything you write is correct, but I think you're refuting something that wasn't intended. I think euank's comment meant that you can't have a concept of "approaches" when you're just dealing with single values. That's what the "just" is intended to mean, you're "just" dealing with numbers, not with sequences. With that reading your comment is misplaced, although I can completely understand the interpretation you put on it, and were that what had been intended, your comment would have been well stated.

by euank 12y ago

pdonis 12y ago

> In fact, the idea of "approaches" cannot happen with just rational numbers. Yes, it can. The sequence 0.9, 0.99, 0.999, ... is entirely composed of rational numbers, and its limit is also a rational number. The issue with rational numbers is that there are sequences which are composed entirely of rational numbers but do not have limits that are rational numbers. An example is 1.4, 1.41, 1.414, ..., i.e., successive truncations of the decimal expansion of sqrt(2). The limit of the sequence is sqrt(2) itself, which is irrational. So in that particular case, the concept of "taking the limit of a sequence" can't be applied within the rational numbers. But that doesn't mean it can't be applied at all; you just need to restrict to sequences that have rational limits.

ColinWright 12y ago
Just a thought ... euank> In fact, the idea of "approaches" cannot euank> happen with just rational numbers. pdonis> Yes, it can. The sequence 0.9, 0.99, 0.999, ... pdonis> is entirely composed of rational numbers, and its pdonis> limit is also a rational number. Everything you write is correct, but I think you're refuting something that wasn't intended. I think euank's comment meant that you can't have a concept of "approaches" when you're just dealing with single values. That's what the "just" is intended to mean, you're "just" dealing with numbers, not with sequences. With that reading your comment is misplaced, although I can completely understand the interpretation you put on it, and were that what had been intended, your comment would have been well stated.